Evaluating a Line Integral in Differential Form In Exercises 53-56, evaluate the line integral along the path C given by x = 2 t , y = 4 t , where 0 ≤ t ≤ 1 . ∫ C ( y − x ) d x + 5 x 2 y 2 d y
Evaluating a Line Integral in Differential Form In Exercises 53-56, evaluate the line integral along the path C given by x = 2 t , y = 4 t , where 0 ≤ t ≤ 1 . ∫ C ( y − x ) d x + 5 x 2 y 2 d y
Solution Summary: The author explains how the line integral displaystyleundersetCint is 258.
Evaluating a Line Integral in Differential Form In Exercises 53-56, evaluate the line integral along the path C given by
x
=
2
t
,
y
=
4
t
,
where
0
≤
t
≤
1
.
∫
C
(
y
−
x
)
d
x
+
5
x
2
y
2
d
y
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Evaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals.
φ(x, y, z) = xy + xz + yz; C: r(t) = ⟨t, 2t, 3t⟩ , for 0 ≤ t ≤ 4
Evaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals.
φ(x, y) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2
Evaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals.
φ(x, y, z) = (x2 + y2 + z2)/2; C: r(t) = ⟨cos t, sin t, t/π⟩ , for 0 ≤ t ≤ 2π
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